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This set of notes focuses on the restriction problem in Fourier analysis. Introduced by Elias Stein in the 1970s, the restriction problem is a key model problem for understanding more general oscillatory integral operators, and which has turned out to be connected to many questions in geometric measure theory, harmonic analysis, combinatorics, number theory, and PDE. Only partial results on the problem are known, but these partial results have already proven to be very useful or influential in many applications.
We work in a Euclidean space ${{\bf R}^d}$. Recall that ${L^p({\bf R}^d)}$ is the space of ${p^{th}}$-power integrable functions ${f: {\bf R}^d \rightarrow {\bf C}}$, quotiented out by almost everywhere equivalence, with the usual modifications when ${p=\infty}$. If ${f \in L^1({\bf R}^d)}$ then the Fourier transform ${\hat f: {\bf R}^d \rightarrow {\bf C}}$ will be defined in this course by the formula

$\displaystyle \hat f(\xi) := \int_{{\bf R}^d} f(x) e^{-2\pi i x \cdot \xi}\ dx. \ \ \ \ \ (1)$

From the dominated convergence theorem we see that ${\hat f}$ is a continuous function; from the Riemann-Lebesgue lemma we see that it goes to zero at infinity. Thus ${\hat f}$ lies in the space ${C_0({\bf R}^d)}$ of continuous functions that go to zero at infinity, which is a subspace of ${L^\infty({\bf R}^d)}$. Indeed, from the triangle inequality it is obvious that

$\displaystyle \|\hat f\|_{L^\infty({\bf R}^d)} \leq \|f\|_{L^1({\bf R}^d)}. \ \ \ \ \ (2)$

If ${f \in L^1({\bf R}^d) \cap L^2({\bf R}^d)}$, then Plancherel’s theorem tells us that we have the identity

$\displaystyle \|\hat f\|_{L^2({\bf R}^d)} = \|f\|_{L^2({\bf R}^d)}. \ \ \ \ \ (3)$

Because of this, there is a unique way to extend the Fourier transform ${f \mapsto \hat f}$ from ${L^1({\bf R}^d) \cap L^2({\bf R}^d)}$ to ${L^2({\bf R}^d)}$, in such a way that it becomes a unitary map from ${L^2({\bf R}^d)}$ to itself. By abuse of notation we continue to denote this extension of the Fourier transform by ${f \mapsto \hat f}$. Strictly speaking, this extension is no longer defined in a pointwise sense by the formula (1) (indeed, the integral on the RHS ceases to be absolutely integrable once ${f}$ leaves ${L^1({\bf R}^d)}$; we will return to the (surprisingly difficult) question of whether pointwise convergence continues to hold (at least in an almost everywhere sense) later in this course, when we discuss Carleson’s theorem. On the other hand, the formula (1) remains valid in the sense of distributions, and in practice most of the identities and inequalities one can show about the Fourier transform of “nice” functions (e.g., functions in ${L^1({\bf R}^d) \cap L^2({\bf R}^d)}$, or in the Schwartz class ${{\mathcal S}({\bf R}^d)}$, or test function class ${C^\infty_c({\bf R}^d)}$) can be extended to functions in “rough” function spaces such as ${L^2({\bf R}^d)}$ by standard limiting arguments.
By (2), (3), and the Riesz-Thorin interpolation theorem, we also obtain the Hausdorff-Young inequality

$\displaystyle \|\hat f\|_{L^{p'}({\bf R}^d)} \leq \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (4)$

for all ${1 \leq p \leq 2}$ and ${f \in L^1({\bf R}^d) \cap L^2({\bf R}^d)}$, where ${2 \leq p' \leq \infty}$ is the dual exponent to ${p}$, defined by the usual formula ${\frac{1}{p} + \frac{1}{p'} = 1}$. (One can improve this inequality by a constant factor, with the optimal constant worked out by Beckner, but the focus in these notes will not be on optimal constants.) As a consequence, the Fourier transform can also be uniquely extended as a continuous linear map from ${L^p({\bf R}^d) \rightarrow L^{p'}({\bf R}^d)}$. (The situation with ${p>2}$ is much worse; see below the fold.)
The restriction problem asks, for a given exponent ${1 \leq p \leq 2}$ and a subset ${S}$ of ${{\bf R}^d}$, whether it is possible to meaningfully restrict the Fourier transform ${\hat f}$ of a function ${f \in L^p({\bf R}^d)}$ to the set ${S}$. If the set ${S}$ has positive Lebesgue measure, then the answer is yes, since ${\hat f}$ lies in ${L^{p'}({\bf R}^d)}$ and therefore has a meaningful restriction to ${S}$ even though functions in ${L^{p'}}$ are only defined up to sets of measure zero. But what if ${S}$ has measure zero? If ${p=1}$, then ${\hat f \in C_0({\bf R}^d)}$ is continuous and therefore can be meaningfully restricted to any set ${S}$. At the other extreme, if ${p=2}$ and ${f}$ is an arbitrary function in ${L^2({\bf R}^d)}$, then by Plancherel’s theorem, ${\hat f}$ is also an arbitrary function in ${L^2({\bf R}^d)}$, and thus has no well-defined restriction to any set ${S}$ of measure zero.
It was observed by Stein (as reported in the Ph.D. thesis of Charlie Fefferman) that for certain measure zero subsets ${S}$ of ${{\bf R}^d}$, such as the sphere ${S^{d-1} := \{ \xi \in {\bf R}^d: |\xi| = 1\}}$, one can obtain meaningful restrictions of the Fourier transforms of functions ${f \in L^p({\bf R}^d)}$ for certain ${p}$ between ${1}$ and ${2}$, thus demonstrating that the Fourier transform of such functions retains more structure than a typical element of ${L^{p'}({\bf R}^d)}$:

Theorem 1 (Preliminary ${L^2}$ restriction theorem) If ${d \geq 2}$ and ${1 \leq p < \frac{4d}{3d+1}}$, then one has the estimate

$\displaystyle \| \hat f \|_{L^2(S^{d-1}, d\sigma)} \lesssim_{d,p} \|f\|_{L^p({\bf R}^d)}$

for all Schwartz functions ${f \in {\mathcal S}({\bf R}^d)}$, where ${d\sigma}$ denotes surface measure on the sphere ${S^{d-1}}$. In particular, the restriction ${\hat f|_S}$ can be meaningfully defined by continuous linear extension to an element of ${L^2(S^{d-1},d\sigma)}$.

Proof: Fix ${d,p,f}$. We expand out

$\displaystyle \| \hat f \|_{L^2(S^{d-1}, d\sigma)}^2 = \int_{S^{d-1}} |\hat f(\xi)|^2\ d\sigma(\xi).$

From (1) and Fubini’s theorem, the right-hand side may be expanded as

$\displaystyle \int_{{\bf R}^d} \int_{{\bf R}^d} f(x) \overline{f}(y) (d\sigma)^\vee(y-x)\ dx dy$

where the inverse Fourier transform ${(d\sigma)^\vee}$ of the measure ${d\sigma}$ is defined by the formula

$\displaystyle (d\sigma)^\vee(x) := \int_{S^{d-1}} e^{2\pi i x \cdot \xi}\ d\sigma(\xi).$

In other words, we have the identity

$\displaystyle \| \hat f \|_{L^2(S^{d-1}, d\sigma)}^2 = \langle f, f * (d\sigma)^\vee \rangle_{L^2({\bf R}^d)}, \ \ \ \ \ (5)$

using the Hermitian inner product ${\langle f, g\rangle_{L^2({\bf R}^d)} := \int_{{\bf R}^d} \overline{f(x)} g(x)\ dx}$. Since the sphere ${S^{d-1}}$ have bounded measure, we have from the triangle inequality that

$\displaystyle (d\sigma)^\vee(x) \lesssim_d 1. \ \ \ \ \ (6)$

Also, from the method of stationary phase (as covered in the previous class 247A), or Bessel function asymptotics, we have the decay

$\displaystyle (d\sigma)^\vee(x) \lesssim_d |x|^{-(d-1)/2} \ \ \ \ \ (7)$

for any ${x \in {\bf R}^d}$ (note that the bound already follows from (6) unless ${|x| \geq 1}$). We remark that the exponent ${-\frac{d-1}{2}}$ here can be seen geometrically from the following considerations. For ${|x|>1}$, the phase ${e^{2\pi i x \cdot \xi}}$ on the sphere is stationary at the two antipodal points ${x/|x|, -x/|x|}$ of the sphere, and constant on the tangent hyperplanes to the sphere at these points. The wavelength of this phase is proportional to ${1/|x|}$, so the phase would be approximately stationary on a cap formed by intersecting the sphere with a ${\sim 1/|x|}$ neighbourhood of the tangent hyperplane to one of the stationary points. As the sphere is tangent to second order at these points, this cap will have diameter ${\sim 1/|x|^{1/2}}$ in the directions of the ${d-1}$-dimensional tangent space, so the cap will have surface measure ${\sim |x|^{-(d-1)/2}}$, which leads to the prediction (7). We combine (6), (7) into the unified estimate

$\displaystyle (d\sigma)^\vee(x) \lesssim_d \langle x\rangle^{-(d-1)/2}, \ \ \ \ \ (8)$

where the “Japanese bracket” ${\langle x\rangle}$ is defined as ${\langle x \rangle := (1+|x|^2)^{1/2}}$. Since ${\langle x \rangle^{-\alpha}}$ lies in ${L^p({\bf R}^d)}$ precisely when ${p > \frac{d}{\alpha}}$, we conclude that

$\displaystyle (d\sigma)^\vee \in L^q({\bf R}^d) \hbox{ iff } q > \frac{d}{(d-1)/2}.$

Applying Young’s convolution inequality, we conclude (after some arithmetic) that

$\displaystyle \| f * (d\sigma)^\vee \|_{L^{p'}({\bf R}^d)} \lesssim_{p,d} \|f\|_{L^p({\bf R}^d)}$

whenever ${1 \leq p < \frac{4d}{3d+1}}$, and the claim now follows from (5) and Hölder’s inequality. $\Box$

Remark 2 By using the Hardy-Littlewood-Sobolev inequality in place of Young’s convolution inequality, one can also establish this result for ${p = \frac{4d}{3d+1}}$.

Motivated by this result, given any Radon measure ${\mu}$ on ${{\bf R}^d}$ and any exponents ${1 \leq p,q \leq \infty}$, we use ${R_\mu(p \rightarrow q)}$ to denote the claim that the restriction estimate

$\displaystyle \| \hat f \|_{L^q({\bf R}^d, \mu)} \lesssim_{d,p,q,\mu} \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (9)$

for all Schwartz functions ${f}$; if ${S}$ is a ${k}$-dimensional submanifold of ${{\bf R}^d}$ (possibly with boundary), we write ${R_S(p \rightarrow q)}$ for ${R_\mu(p \rightarrow q)}$ where ${\mu}$ is the ${k}$-dimensional surface measure on ${S}$. Thus, for instance, we trivially always have ${R_S(1 \rightarrow \infty)}$, while Theorem 1 asserts that ${R_{S^{d-1}}(p \rightarrow 2)}$ holds whenever ${1 \leq p < \frac{4d}{3d+1}}$. We will not give a comprehensive survey of restriction theory in these notes, but instead focus on some model results that showcase some of the basic techniques in the field. (I have a more detailed survey on this topic from 2003, but it is somewhat out of date.)
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This is a well-known problem in multilinear harmonic analysis; it is fascinating to me because it lies barely beyond the reach of the best technology we have for these problems (namely, multiscale time-frequency analysis), and because the most recent developments in quadratic Fourier analysis seem likely to shed some light on this problem.

Recall that the Hilbert transform is defined on test functions $f \in {\mathcal S}({\Bbb R})$ (up to irrelevant constants) as

$Hf(x) := p.v. \int_{\Bbb R} f(x+t) \frac{dt}{t},$

where the integral is evaluated in the principal value sense (removing the region $|t| < \epsilon$ to ensure integrability, and then taking the limit as $\epsilon \to 0$.)